what is the criterion for a random variable(continous) for existence of probability density function for it? Could you provide some cases of random variable(continous) where pdf ceases to exist.

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    $\begingroup$ How are you defining a continuous random variable? $\endgroup$ – M. Vinay Jun 19 '14 at 5:17
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    $\begingroup$ Sometimes "continuous random variable" means "random variable with a density function." $\endgroup$ – Qiaochu Yuan Jun 19 '14 at 5:18
  • $\begingroup$ a continous random variable has continous CDF function. $\endgroup$ – VKV Jun 19 '14 at 5:18
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    $\begingroup$ It happens if the cdf is absolutely continuous. $\endgroup$ – André Nicolas Jun 19 '14 at 5:20
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    $\begingroup$ An example in which it is not absolutely continuous is en.wikipedia.org/wiki/Cantor_distribution $\endgroup$ – M. Vinay Jun 19 '14 at 5:22

A (real-valued) random variable $X$ has density $f$ if you can write $$P(X \leq x) = \int_{-\infty}^x f(y) dy$$.

For a random variable which does not admit a density, take any discrete random variable - geometric, bernoulli, etc.

  • $\begingroup$ i want to know whether every continous random variable has a density function or not. thanks $\endgroup$ – VKV Jun 19 '14 at 5:13
  • $\begingroup$ This answer is off-topic, the OP is basically asking about random variables of the third type, neither absolutely continuous nor discrete. $\endgroup$ – Did Jun 19 '14 at 5:51
  • $\begingroup$ This answer was on topic for the original question. $\endgroup$ – Batman Jun 20 '14 at 2:50

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